Subgroups — #20293024

Order 6848.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16812108	C₂	1	2	yes	maximal
2	#16812041	C₂	1	1	yes	maximal, characteristic
3	#16812040	C₂	1	2	yes	maximal
4	C₈×Dic₁₀₇	C₂	1	1	yes	maximal, characteristic
5	#106×C₁₀₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₁₄	—	2	2	no	—
7	C₂×C₁₀₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₄⋊Dic₁₀₇	C₂²	1	2	yes	—
9	Dic₁₀₇⋊C₄	—	2	2	no	—
10	C₄×Dic₁₀₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₁₀₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₄×C₁₀₇	C₂²	1	2	yes	—
13	C₂×C₄×C₁₀₇	—	2	2	no	—
14	C₂×C₄×C₁₀₇	C₂³	1	1	yes	characteristic
15	C₂×Dic₁₀₇	—	2	2	no	—
16	C₂×Dic₁₀₇	D₄	1	2	yes	—
17	Dic₂₁₄	—	4	4	no	—
18	C₈×C₁₀₇	Q₈	1	2	yes	complemented
19	C₁₀₇⋊C₈	Q₈	1	2	yes	complemented
20	C₂²×C₁₀₇	C₂×D₄	1	1	yes	characteristic
21	C₄×C₁₀₇	—	4	2	no	—
22	C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
23	C₄×C₁₀₇	C₂×Q₈	1	1	yes	characteristic
24	Dic₁₀₇	—	4	2	no	—
25	Dic₁₀₇	SD₁₆	1	4	yes	—
26	C₂×C₁₀₇	C₄⋊Q₈	1	1	yes	characteristic
27	C₂×C₁₀₇	C₂×SD₁₆	1	2	yes	—
28	C₁₀₇	#498	1	1	yes	characteristic, complemented, Sylow
29	#498	—	107	1	no	maximal, Sylow
30	C₄⋊Q₈	—	107	2	no	—
31	#106	—	107	1	no	—
32	#106	—	107	2	no	—
33	#106	D₁₀₇	1	1	yes	characteristic
34	C₄×C₈	—	107	1	no	—
35	C₂×Q₈	—	214	2	no	—
36	C₂×C₈	—	107	1	no	—
37	C₂×C₈	C₂×D₁₀₇	1	1	yes	characteristic
38	C₄⋊C₄	—	107	2	no	—
39	C₄⋊C₄	—	214	2	no	—
40	C₄⋊C₄	C₂×D₁₀₇	1	2	yes	—
41	C₄²	—	107	1	no	—
42	Q₈	—	428	4	no	—
43	C₂×C₄	—	2	2	no	—
44	C₂×C₄	—	107	2	no	—
45	C₂×C₄	—	214	2	no	—
46	C₂×C₄	C₂²×D₁₀₇	1	1	yes	characteristic
47	C₈	—	107	2	no	—
48	C₈	Dic₂₁₄	1	2	yes	complemented
49	C₂²	D₄×D₁₀₇	1	1	yes	characteristic
50	C₄	—	4	2	no	—
51	C₄	—	107	4	no	—
52	C₄	—	428	2	no	—
53	C₄	Q₈×D₁₀₇	1	1	yes	characteristic
54	C₄	C₂×Dic₂₁₄	1	1	yes	characteristic
55	C₂	#16812108	1	1	yes	characteristic
56	C₂	SD₁₆×D₁₀₇	1	2	yes	—

	subgroup	G/H
center	C₂²	D₄×D₁₀₇
commutator	C₂×C₄×C₁₀₇	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₇
Fitting	#106×C₁₀₇	C₂
socle	C₂²×C₁₀₇	C₂×D₄
hypercenter	#106	D₁₀₇
hypocenter	C₁₀₇	#498
LCS	G ▷ C₂×C₄×C₁₀₇ ▷ C₂×C₁₀₇ ▷ C₁₀₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #106

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
18. C₈×C₁₀₇	Q₈	semidirect
19. C₁₀₇⋊C₈	Q₈	semidirect
28. C₁₀₇	#498	semidirect
48. C₈	Dic₂₁₄	semidirect